Lectures_on_Cuves_Fitting

8/7/2019 Lectures_on_Cuves_Fitting

1/36

LECTURE - 7


2/36

LEAST SQUARES

CURVE FITTING


3/36

Motivation

Given a set of experimental data

x 1 2 3

y 5.1 5.9 6.3

The relationship between x

and y may not be clearwe want to find an

expression for f(x)1 2 3


4/36

KEPPLER THIRD LAW OF

PLANETARY MOTION


5/36


6/36

Curve Fitting

Given a set of

tabulated data, find acurve or a function

that best representsthe data.


7/36

EXPERIMENTAL ERROR

xi x1 x2 . xn

yi y1 y2 . yn

Given

The form of the function is assumed to

be known but the coefficients are

unknown.

kkk eyxf!

)(The difference is assumed to be the result of

experimental error


8/36

ERROR

2

1

1

2

2

11

k

k

N

1kk

))(1

((f)EError-RMS

)(1)(EErrorAverage

)(max)(EErrorMaximum

N.k1for)(e)deviationscalled(alsoerrorcalledis

datarecordedtheand)f(xvaluetrueebetween th

differenceThedata.ofsetthe)}y,{(x

!

!

!

ee!

g

N

kk

N

kk

kk

kk

yxfN

yxfN

f

yxff

yxf

beLet


9/36

LEAST- SQUARES LINE

INTRODUCTION


10/36

LEAST- SQUARES LINE

The least square liney=f(x)=Ax+B is the line

that minimizes RMS-error


11/36

Please write down the black boardsummary

FINDING LEAST-SQUARES LINE


12/36

Determine the Unknowns

?),(minimizetoandobtainedoHo

)(),(

minimizetoba,indant toWe

0

2

baEba

bxafbaEN

k

kk !!


13/36


?),(minimizetoandobtainwedoow

)(),(

minimizetoba,findwant toWe

0

2

baba

bxafban

i

ii

*

!* !


14/36


0

),(

0),(

minimumfor theconditionNecessary

!x

x

!x

x

b

ba

aba


15/36

Remember

!xx

!

!!

!!

n

kk

n

ki

n

kk

n

kk

xgaxga

axadx

d

11

11

)()(


16/36

Example 1

!

!

!!x

x

!!x

x

!!!

!!

!

!

N

kkk

N

kk

N

kk

N

kk

N

kk

k

N

kkk

N

kkk

yxbxax

ybxaN

xybxabbaE

ybxaa

baE

11

2

1

11

1

1

EquationsNormal

20),(

20),(


17/36

Example 1

!

!

!!

!!

!!!

N

k

k

N

k

k

N

k

k

N

k

k

N

k

k

N

k

k

N

k

kk

xbyN

a

xxN

yxyxN

b

11

2

11

2

111

1

givesEquationsNormaltheolving


18/36

Example 1

k 1 2 3 sum

xk 1 2 3 6

yk 5.1 5.9 6.3 17.3

xk2 1 4 9 14

xk yk 5.1 11.8 18.9 35.8


19/36

Example 1

60.04.5667

8.35146

3.1763

EquationsNormal

11

2

1

11

!!

!

!

!

!

!!!

!!

baSolving

ba

ba

yxbxax

ybxaN

N

k

kk

N

k

k

N

k

k

N

k

k

N

k

k


20/36

Power Fit

AxY M!


21/36

Example 2

data.fit theto

)cos()ln()(

formtheoffunctionafindtorequiredisIt

xecxbxaxf !

x 0.24 0.65 0.95 1.24 1.73 2.01 2.23 2.52

y 0.23 -0.23 -1.1 -0.45 0.27 0.1 -0.29 0.24


22/36

Example 2

EquationsNormal

c

cbab

cba

acba

!x

*x

!

x

*x

!x*x

0),,(

0),,(

0),,(

minimumfor theconditionNecessary


23/36

Example 2

equationsnormalthesolveandsumstheEvaluate

)()())((cos)()(ln

)(cos)()(cos)(cos)(cos)(ln

)(ln)()(ln)(cos)(ln)(ln

8

1

8

1

28

1

8

1

8

1

8

1

8

1

28

1

8

1

8

1

8

1

28

1

kkkk

k

k

x

k kk

x

k

x

k

x

k k

k

kk

x

k

k

k

kk

k

k

k

kk

x

k

kk

k

k

k

k

eyecexbexa

xyexcxbxxa

xyexcxxbxa

!!!!

!!!!

!!!!

!

!

!


24/36

How do you judge performance?

best?select theyoudoow

data,fit thetofunctionsmoreorGiven two

sense.square

leastin thebesttheis)(smallerinresulting

functionTheone.eachfor)(computethen

functioneachforparameterstheDetermine

:

2

2

fE

fE

Answer


25/36

Multiple Regression

Example:

Given the following data

It is required to determine afunction of two variables

f(x,t) = a + b x + c t

to explain the data that is bestin the least square sense.

t 0 1 2 3

x 0.1 0.4 0.2 0.2

f(x,t) 3 2 1 2


26/36

SolutionofMultiple Regression

Construct , the sumof the square of theerror and derive the

necessary conditionsby equating thepartial derivativeswith respect to theunknown parameters

to zero then solvethe equations.

* t 0 1 2 3

x 0.1 0.4 0.2 0.2

f(x,t) 3 2 1 2


27/36

SolutionofMultiple Regression

02),,(

02

),,(

02),,(conditionsNecessary

),,(

),(

4

1

4

1

4

1

4

1

2

!!x

*x

!!x

*x

!!x

*x

!*

!

!

!

!

!

i

i

iii

ii

iii

i

iii

i

iii

tfctbxac

cba

xfctbxab

cba

fctbxaa

cba

fctbxacba

ctbxatxf


28/36

Nonlinear least squares problems

EXAMPLES


29/36

data.fit thebestthatformtheoffuctionafind

bx

ae

ii

ii

i

bx

i

ibx

bx

i

ibx

i

ibx

ebayaeb

eyaea

yae

!

!

!

!!

x

*x

!!

x*x

!*

3

1

3

1

3

1

2

0

0

usingobtainedareEquationsNormal

Nonlinear Problem

Givenx 1 2 3

y 2.4 5. 9


30/36

data.fit thebestthatformtheoffuctionafind

bx

ae

solve)easier to(useillWe

usingoInstead

)ln()ln(

bxln(a)ln(y)zDe ine

3

1

2

3

1

2

!

!

!*

!*

!!

!!

i

ii

ii

bx

ii

zbx

yae

yzandaet

i

E

E

Alternative Solution(LinearizationMethod)

Givenx 1 2 3

y 2.4 5. 9


31/36

InconsistentSystem ofEquations

solutionNo

EquationsofsystemntinconsisteisThis

10

6

4

1.43

2221

equationsofsystemfollowingtheSolve

:Problem

2

1

!

x

x


32/36

InconsistentSystem ofEquationsReasons

Inconsistent equations

may occur because of

errors in formulatingthe problem, errors in

collecting the data or

computational errors.

Solution if all lines intersect

at one point


33/36

InconsistentSystem ofEquationsFormulationasa leastsquares problem

errorsquaresleasttheminimizetoandFind

10

6

4

1.43

22

21

asequationsthecan viewWe

21

3

2

1

2

1

xx

x

x

!

I

I

I


34/36

Solution

12262.4936.6

)822416()62.3388()6.2484(0

)011.43(2.86)22(44)2(40

926.3628

)60248()6.2484()1882(0

)011.43(66)22(44)2(20

minimizetoandFind

)011.43(6)22(4)2(

21

21

2121212

21

21

2121211

21

221

221

221

!

!

!!x*x

!

!

!!x

*x

*

!*

xx

xx

xxxxxxx

xx

xx

xxxxxxx

xx

xxxxxx


35/36

Solution

0.9799,2.0048

:

12262.4936.6

926.3628

:equationsNormal

21

21

21

!!

!

!

xx

Solution

xx

xx


36/36

data.it thebestthatb)/(ax1ormtheouctionaind

!

!

!*

!*

!

!!

!

3

1

2

3

1

2

usewillWe

bax

1usingoInstead

1

bax1

ze ine

bax

1f(x)

i

ii

i

i

ii

zbax

y

yzLet

y

Examples(LinearizationMethod)

Givenx 1 2 3

y 0.23 .2 .14

Documents

Lectures_on_Cuves_Fitting